This article addresses Taruek’s much discussed Number Problem from a non-consequentialist point of view. I argue that some versions of the Number Problem have no solution, meaning that no alternative is at least as choice-worthy as the others, and that the best way to behave in light of such moral indeterminacy is to let chance make the decision. I contrast my proposal with F M Kamm ’s nonconsequentialist argument for saving the greatest number, the Argument for Best Outcomes, (...) which I argue does not follow from the premises it is based on. (shrink)

You must either save a group of m people or a group of n people. If there are no morally relevant diff erences among the people, which group should you save? is problem is known as the number problem. e recent discussion has focussed on three proposals: (i) Save the greatest number of people, (ii) Toss a fair coin, or (iii) Set up a weighted lottery, in which the probability of saving m people is m / m + (...) n , and the probability of saving n people is n / m + n . is contribution examines a fourth alternative, the mixed solution, according to which both fairness and the total number of people saved count. It is shown that the mixed solution can be defended without assuming the possibility of interpersonal comparisons of value. (shrink)

In this article I criticize the non-consequentialist Weighted Lottery (WL) solution to the choice between saving a smaller or a larger group of people. WL aims to avoid what non-consequentialists see as consequentialism's unfair aggregation by giving equal consideration to each individual's claim to be rescued. In so doing, I argue, WL runs into another common objection to consequentialism: it is excessively demanding. WL links the right action with the outcome of a fairly weighted lottery, which means that an agent (...) can only act rightly if s/he has actually run the lottery. In many actual cases, this involves epistemic demands that can be almost impossible to meet. I argue that plausible moral principles cannot make such extreme epistemic demands. (shrink)

This chapter presents a historical analysis of pseudoscience, tracking down the coinage and currency of the term and explaining its shifting meaning in tandem with the emerging historical identity of science. The discussions cover the invention of pseudoscience; science and pseudoscience in the late nineteenth century; pseudoscience in the new century; and pseudoscience and its critics in the late twentieth century.

This article develops an unconventional perspective on the utilitarianism of Bentham and Mill in at least four areas. First, it is shown that both authors conceived of utility as irreducibly multi-dimensional, and that Bentham in particular was very much aware of the ambiguity that multi-dimensionality imposes upon optimal choice under the greatest happiness principle. Secondly, I argue that any attribution of intrinsic worth to any form of human behaviour violates the first principles of Bentham's and Mill's utilitarianism, and that this (...) renders both authors immune to the claim by G. E. Moore that they committed a ‘naturalistic fallacy’. Thirdly, in light of these contentions, I find no flaw in Mill's ‘proof of utility’. Fourthly, I use the notion of intrapersonal utility weights to provide an interpretation of Mill's qualitative hedonism that is entirely consistent with his value monism. (shrink)

Plato believed that behind everything in the universe lie mathematical principles. Plato was inspired by Pythagoras (571 BCE), who developed a school of mathematics at Crotona that studied sacred geometry as a form of religion. The school’s motto was “God is number,” or “All is Number”. Pythagoras believed that numbers represented God in pattern, symmetry, and infinity. When one of its students, Hippasus told the world the secret of the existence of irrational numbers, Greek geometry was born and (...) Pythagoras’ idea of divinity in numbers died because how could God not be perfect and symmetrical? In Plato’s Republic he discusses something called The Divided Line, which is a map, of sorts, for reaching what he calls the highest Good, which is the ultimate truth where one realizes the true state of the universe and can see the world for what it really is. Many mathematicians have attempted to plot Plato’s Divided Line only to come across a litany of problems and conundrums. Some have said that it the Divided Line cannot be plotted and is merely an allegory not meant to be plotted. This paper discusses some of the conundrums preventing the plotting of Plato’s Divided Line (not an exhaustive list), including Whole ‘vs’ Separate, Equality ‘vs’ Ontological Dissimilarity, Linear ‘vs’ Non-linear, and Infinity ‘vs’ Finite. This paper also explores a new understanding of the Allegory of the Cave in light of ‘the problem of the irrational.’ In exploring the link between the Divided Line and the ‘the problem of the irrational,’ I was able to plot it. It was found that the Divided Line is not a line in the linear sense, but a spiral, the Golden Ratio! This paper is an example of a new category of scholarly inquiry I call “Math Theory” based on scholarly mathematical axioms in theory, rather than including actual maths. In my papers I use existing mathematical equations and place them in an encompassing theory, rather than finding new formulae to fit an existing theory. Keywords: Pythagoras, divided line, math theory, highest good, all is number. (shrink)

Discussion of the “problem of numbers” in morality has focused almost exclusively on the moral significance of numbers in whom-to-rescue cases: when you can save either of two groups of people, but not both, does the number of people in each group matter morally? I suggest that insufficient attention has been paid to the moral significance of numbers in other types of case. According to common-sense morality, numbers make a difference in cases, like the famous (...) Trolley Case, where we must choose whether to kill a person (or persons) as a side effect of saving a greater number. I argue that recognition of the role of numbers in killing cases forces us to reassess purported solutions to the problem of numbers. (shrink)

Those inclined to believe in the existence of propositions as traditionally conceived might seek to reduce them to some other type of entity. However, parsimonious propositionalists of this type are confronted with a choice of competing candidates – for example, sets of possible worlds, and various neo-Russellian and neo-Fregean constructions. It is argued that this choice is an arbitrary one, and that it closely resembles the type of problematic choice that, as Benacerraf pointed out, bedevils the attempt to reduce (...) class='Hi'>numbers to sets – should the number 2 be identified with the set Ø or with the set Ø, Ø? An “argument from arbitrary identification” is formulated with the conclusion that propositions (and perhaps numbers) cannot be reduced away. Various responses to this argument are considered, but ultimately rejected. The paper concludes that the argument is sound: propositions, at least, are sui generis entities. (shrink)

This chapter presents arguments for two slightly different versions of the thesis that the value of persons is incomparable. Both arguments allege an incompatibility between the demands of a certain kind of practical reasoning and the presuppositions of value comparisons. The significance of these claims is assessed in the context of the “Numbers problem”—the question of whether one morally ought to benefit one group of potential aid recipients rather than another simply because they are greater in number. It (...) is argued that many of the popular approaches to this problem—even ones that avoid the aggregation of personal value—are imperiled by the incomparability theses. -/- . (shrink)

Human perception responds primarily to sound character rather than sound level. Wind farms are unique sound sources and exhibit special audible and inaudible characteristics that can be described as modulating sound or as a tonal complex. Wind farm compliance measures based on a specified noise number alone will fail to address problems with noise nuisance. The character of wind farm sound, noise emissions from wind farms, noise prediction at residences, and systemic failures in assessment processes are examined. Human perception of (...) wind farm sound is compared with noise assessment measures and complaint histories. The adverse effects on health of persons susceptible to noise from wind farms are examined and a hypothesis, the concept of heightened noise zones (pressure variations), as a marker for cause and effect is advanced. A sound level of LAeq 32 dB outside a residence and above an individual’s threshold of hearing inside the home are identified as markers for serious adverse health effects affecting susceptible individuals. The article is referenced to the author’s research, measurements, and observations at different wind farms in New Zealand and Victoria, Australia. (shrink)

This paper aims to show that Selim Berker’s widely discussed prime number case is merely an instance of the well-known generality problem for process reliabilism and thus arguably not as interesting a case as one might have thought. Initially, Berker’s case is introduced and interpreted. Then the most recent response to the case from the literature is presented. Eventually, it is argued that Berker’s case is nothing but a straightforward consequence of the generality problem, i.e., the problematic aspect (...) of the case for process reliabilism (if any) is already captured by the generality problem. (shrink)

Reliabilism suffers from a problem with long sequences of justifications. The theory of justification provided in process reliabilism allows for an implausibly large extension of ‘justified belief’. According to process reliabilist theory, it is possible that a justifying cognitive process has an arbitrarily low probability of being successful and a justified belief an arbitrarily low probability of being true. This result violates reliabilism’s aims as well as our ordinary standards of justification.

Simple-type theory is widely regarded as inadequate to capture the metaphysics of mathematics. The problem, however, is not that some kinds of structure cannot be studied within simple-type theory. Even structures that violate simple-types are isomorphic to structures that can be studied in simple-type theory. In disputes over the logicist foundations of mathematics, the central issue concerns the problem that simple-type theory fails to assure an infinity of natural numbers as objects. This paper argues that the (...) class='Hi'>problem of infinity is based on a metaphysical prejudice in favor of numbers as objects — a prejudice that mathematics can get along without. (shrink)

Diogenes Laërtius and Plinius report that Anaximander made a globe, meaning a celestial globe. These statements must be due to an anachronistic misunderstanding, as a celestial globe presupposes a conception of a spherical universe in which the stars make up the outermost sphere. According to Anaximander, however, the stars are nearest to the earth, as is confi rmed by Aëtius and Hippolytus. Generally speaking, Anaximander’s universe of a column-drum-like earth at the center of the concentric wheels of the celestial bodies (...) can hardly be called spherical. Nevertheless we may assume that Anaximander, who drew a map of the earth, offered some representation of his cosmological ideas as well. Although a three-dimensional representation like a celestial globe is not likely, one may readily imagine that he drew a two-dimensional picture, a kind of ground-plan or map of his conception of the cosmos. The doxographers will erroneously have taken an account about this map as standing for a celestial globe such as they knew from their own experience. In this article I will deal with the problems that arise when we try to draw a map of Anaximander’s universe, and suggest ways to solve them. In discussing several authors who have studied the subject I will point out some bothersome inconsistencies and mistakes in their renditions. One problem, however, which has not been noticed before, will prove to be insolvable within the context of the doxographical evidence. I will argue that the only way to meet this problem is to look upon it as circumstantial evidence for the supposition that Anaximander never made a three-dimensional model whatsoever of his conception of the universe. (shrink)

Suppose we can save either a larger group of persons or a distinct, smaller group from some harm. Many people think that, all else equal, we ought to save the greater number. This article defends this view (with qualifications). But unlike earlier theories, it does not rely on the idea that several people's interests or claims receive greater aggregate weight. The argument starts from the idea that due to their stakes, the affected people have claims to have a say in (...) the rescue decision. As rescuers, our primary duty is to respect these procedural claims, which we must do by doing what these people would decide, in a process where each is given an equal vote on the matter. So in cases where each votes in their own self-interest, respect for their equal right to decide, or their autonomy, will lead us to save the greater number. The argument is explained in detail, with special attention to the questions of how, exactly, it avoids aggregation, and of why majority rule is superior to lottery procedures. The view has further advantages. Especially, it explains the “partial” relevance of numbers in cases involving unequal harms, and it does so in a way that dissolves the appearance of paradox that besets theories of “partial aggregation.”. (shrink)

Hofweber (Ontology and the ambitions of metaphysics, Oxford University Press, 2016) argues for a thesis he calls “internalism” with respect to natural number discourse: no expressions purporting to refer to natural numbers in fact refer, and no apparent quantification over natural numbers actually involves quantification over natural numbers as objects. He argues that while internalism leaves open the question of whether other kinds of abstracta exist, it precludes the existence of natural numbers, thus establishing what he (...) calls “restricted nominalism” about natural numbers. We argue that Hofweber’s internalism fails to establish restricted nominalism. Not only is his primary argument for restricted nominalism invalid, the analysis of quantification proposed threatens to collapse internalism into either a traditional form of error theory or realism. (shrink)

Este artigo trata do chamado “problema de incompatibilidade de cores”, enfrentado por Wittgenstein no Tractatus Logico-Philosophicus, em seus Manuscritos inéditos 105 e 106, e em seu único artigo publicado – “Some Remarks on Logical Forms” (SRLF). Nossa tarefa será dupla. Primeiro, pretendemos mostrar como e por que Wittgenstein ficou preso nesse dilema. Nossa segunda tarefa será muito mais específica.Tentaremos elucidar alguns detalhes sobre a impossibilidade de reduzir predicados de cor a unidades mais fundamentais de brilho, croma e intensidade. Nosso objetivo (...) é mostrar como e por que os números tiveram que ser introduzidos na estrutura interna das proposições elementares. Também queremos investigar como Wittgenstein tentou fazer isso através da ideia de um “esquema” e uma operação nãover-funcional. Pretendemos também apresentar algumas vantagens que Wittgenstein obteve ao usar uma notação de barra para sua construção dos “esquemas”.AbstractThis paper deals with the so-called “color incompatibility problem”, faced by Wittgenstein in the Tractatus Logico-Philosophicus, in his unpublished Manuscripts 105 and 106, and in his only published paper – “Some Remarks on Logical Forms” – (SRLF, for now on). Our task will be a twofold one. First, we aspire to show how and why Wittgenstein got stuck into that dilemma. Our second task will be much more specific. We will try to elucidate some details about the impossibility of reducing color predicates to more fundamental units of brightness, chroma, and intensity. We aim to show how and why numbers had to be introduced into the inner structure of elementary propositions. We also want to investigate how he tried to do that through the idea of a “scheme” and a non-truth functional operation. We aim also to present some advantages Wittgenstein gained from using a bar notation for his construal of the “schemes”. (shrink)

We introduce and study certain classes of optimization problems over the real numbers. The classes are defined by logical means, relying on metafinite model theory for so called R-structures . More precisely, based on a real analogue of Fagin's theorem [12] we deal with two classes MAX-NPR and MIN-NPR of maximization and minimization problems, respectively, and figure out their intrinsic logical structure. It is proven that MAX-NPR decomposes into four natural subclasses, whereas MIN-NPR decomposes into two. This gives a (...) real number analogue of a result by Kolaitis and Thakur [13] in the Turing model. Our proofs mainly use techniques from [17]. Finally, approximation issues are briefly discussed. (shrink)

The main question addressed in this essay is whether quarks have been observed in any sense and, if so, what might be meant by this use of the term, observation. In the first (or introductory) section of the paper, I explain that well-known researchers are divided on the answers to these important questions. In the second section, I investigate microphysical observation in general. Here I argue that Wilson's analogy between observation by means of high-energy accelerators and observation by means of (...) microscopes is misleading, for at least three reasons. Moreover, so long as high-energy observation is accomplished by means of spark or bubble chambers, then sentences about these observations do not meet Maxwell's criterion, that observation statements are quickly decidable. I argue, however, that this criterion is not a good norm for what is observable in high-energy physics, both because it would result in our describing a great many allegedly observed particle events as unobserved or theoretical, and because it fails to distinguish the reasons why some observation statements might not be quickly decidable. Most important, Maxwell's criterion fails because, contrary to Hanson's analysis, it presupposes that seeing does not involve both seeing as and seeing that.With this background concerning what is meant by general microphysical observation, in the third section of the essay, I discuss what might be meant by a more particular type of observation, that of the quark via scattering events. I employ Feinberg's distinction concerning observation of manifest, versus existent, particles and claim that the alleged indirect observation of quarks as existent particles is really based on a retroductive inference. I explain which premise in the retroductive argument appears most open to the charge of being theoretical (in a damaging sense) and less substantiated by observation. (shrink)

You ought to save a larger group of people rather than a distinct smaller group of people, all else equal. A consequentialist may say that you ought to do so because this produces the most good. If a non-consequentialist rejects this explanation, what alternative can he or she give? This essay defends the following explanation, as a solution to the so-called numbers problem. Its two parts can be roughly summarised as follows. First, you are morally required to want (...) the survival of each stranger for its own sake. Secondly, you are rationally required to achieve as many of these ends as possible, if you have these ends. (shrink)

This paper discusses the "numbers problem," the problem of explaining why you should save more people rather than fewer when forced to choose. Existing non-consequentialist approaches to the problem appeal to fairness to explain why. I argue that this is a mistake and that we can give a more satisfying answer by appealing to requirements of charity or beneficence.

Experimental research by social and cognitive psychologists has established that cooperative groups solve a wide range of problems better than individuals. Cooperative problem solving groups of scientific researchers, auditors, financial analysts, air crash investigators, and forensic art experts are increasingly important in our complex and interdependent society. This comprehensive textbook--the first of its kind in decades--presents important theories and experimental research about group problem solving. The book focuses on tasks that have demonstrably correct solutions within mathematical, logical, scientific, (...) or verbal systems, including algebra problems, analogies, vocabulary, and logical reasoning problems.The book explores basic concepts in group problem solving, social combination models, group memory, group ability and world knowledge tasks, rule induction problems, letters-to-numbers problems, evidence for positive group-to-individual transfer, and social choice theory. The conclusion proposes ten generalizations that are supported by the theory and research on group problem solving. Group Problem Solving is an essential resource for decision-making research in social and cognitive psychology, but also extremely relevant to multidisciplinary and multicultural problem-solving teams in organizational behavior, business administration, management, and behavioral economics. (shrink)

Over the last decades, understanding the sources of the difficulty of Bayesian problem solving has been an important research goal, with the effects of numerical format and individual numeracy being widely studied. However, the focus on the comprehension of probability numbers has overshadowed the relational reasoning demand of the Bayesian task. This is particularly the case when the statistical data are verbally described since the requested quantitative relation (posterior ratio) is misaligned with the presented ones (prior and likelihood (...) ratios). In this regard, here I develop the proposal that research on Bayesian reasoning might improve by considering the notational alignment framework of mathematical problem-solving. Specifically, this framework can help to understand the sources of the main difficulties underlying Bayesian inferences based on verbal descriptions. In essence, the present proposal supports the general claim in math education regarding the need to foster relational comprehension to avoid misleading alignments and improve problem solving. (shrink)

In “How Should We Aggregate Competing Claims?” Alex Voorhoeve develops a theory—Aggregate Relevant Claims (ARC)—which aims to reconcile intuitive judgments for and against aggregating claims in different situations. I argue that ARC does not justify these intuitions but instead ultimately relies on them. We ought not to trust the intuition in favor of nonaggregation, so we ought not to trust ARC. I then show that the nonaggregative part of ARC has a number of unacceptable implications. These problems afflict all nonaggregative (...) theories. Finally, I present a positive argument for full-blooded aggregation. The numbers always count. (shrink)

In recent years, many non-consequentialists such as Frances Kamm and Thomas Scanlon have been puzzling over what has come to be known as the Number Problem, which is how to show that the greater number in a rescue situation should be saved without aggregating the claims of the many, a typical kind of consequentialist move that seems to violate the separateness of persons. In this article, I argue that these non-consequentialists may be making the task more difficult than necessary, (...) because allowing aggregation does not prevent one from being a non-consequentialist. I shall explain how a non-consequentialist can still respect the separateness of persons while allowing for aggregation. (shrink)

Pairs of sentences like the following pose a problem for ontology: (1) Jupiter has four moons. (2) The number of moons of Jupiter is four. (2) is intuitively a trivial paraphrase of (1). And yet while (1) seems ontologically innocent, (2) appears to imply the existence of numbers. Thomas Hofweber proposes that we can resolve the puzzle by recognizing that sentence (2) is syntactically derived from, and has the same meaning as, sentence (1). Despite appearances, the expressions ‘the (...) number of moons of Jupiter’ and ‘four’ do not function semantically as singular terms in (2). Hofweber’s primary evidence for this proposal concerns differences in the focus-related communicative functions of (1) and (2). In this paper I raise several serious problems for Hofweber’s proposal, and for his attempt to support it by appeal to focus-related phenomena. I conclude by offering independent evidence for an alternative, purely pragmatic resolution of the ontological puzzle. (shrink)

Dove and Machery both argue that recent findings about the nature of numerical representation present problems for Concept Empiricism. I shall argue that, whilst this evidence does challenge certain versions of CE, such as Prinz, it needn’t be seen as problematic to the general CE approach. Recent research can arguably be seen to support a CE account of number concepts. Neurological and behavioral evidence suggests that systems involved in the perception of numerical properties are also implicated in numerical cognition. Furthermore, (...) the discovery of associations between spatial and numerical representations also lends independent support to a CE approach. Although these findings support CE in general, certain versions of the theory may need revising in order to accommodate them. In particular, it may be necessary to either jettison Prinz's Modal Specificity Hypothesis or to revise one’s method for individuating modal representational formats. (shrink)